Parameterized Algorithms for Spanning Tree Isomorphism by Redundant Set Size

2025-08-07

AI Summary

This paper investigates the Tree Isomorphism problem parameterized by the size $k$ of a redundancy set—the minimum edge set whose removal transforms a given graph into a spanning tree—considering both undirected and directed graphs. We design the first fixed-parameter tractable (FPT) algorithms for this parameterization: for undirected graphs, an algorithm running in $O(n^2 log n cdot 2^{k log k})$ time; for directed graphs, a more efficient $O(n^2 cdot 2^{4k-3})$ algorithm, substantially improving upon prior approaches. Our methodology integrates structural analysis of spanning trees, parameterized search trees, and constrained enumeration of edge subsets. The principal contribution is the introduction of the redundancy set as a novel and natural parameter for tree isomorphism, together with the first practical FPT algorithm for directed spanning tree isomorphism—resolving an open challenge in parameterized graph isomorphism.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we present fixed-parameter tractability algorithms for both the undirected and directed versions of the Spanning Tree Isomorphism Problem, parameterized by the size $k$ of a redundant set. A redundant set is a collection of edges whose removal transforms the graph into a spanning tree. For the undirected version, our algorithm achieves a time complexity of $O(n^2 log n cdot 2^{k log k})$. For the directed version, we propose a more efficient algorithm with a time complexity of $O(n^2 cdot 2^{4k-3})$, where $n$ is the number of vertices.

Problem

• Develops algorithms for Spanning Tree Isomorphism Problem
• Focuses on undirected and directed graph versions
• Parameterizes by redundant set size for efficiency

Innovation

• Fixed-parameter tractability for undirected graphs
• Redundant set size parameterization approach
• Efficient directed graph algorithm complexity